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The General Linear Model
Christophe Phillips
Cyclotron Research Centre
University of Liège, Belgium
SPM Short Course
London, May 2011
Normalisation
Statistical Parametric MapImage time-series
Parameter estimates
General Linear ModelRealignment Smoothing
Design matrix
Anatomicalreference
Spatial filter
StatisticalInference
RFT
p <0.05
Passive word listeningversus rest
7 cycles of rest and listening
Blocks of 6 scanswith 7 sec TR
Question: Is there a change in the BOLD response between listening and rest?
Stimulus function
One session
A very simple fMRI experiment
stimulus function
1. Decompose data into effects and error
2. Form statistic using estimates of effects and error
Make inferences about effects of interest
Why?
How?
datastatisti
c
Modelling the measured data
linearmode
l
effects estimate
error estimate
Time
BOLD signal
Tim
e
single voxeltime series
single voxeltime series
Voxel-wise time series analysis
Modelspecification
Modelspecification
Parameterestimation
Parameterestimation
HypothesisHypothesis
StatisticStatistic
SPMSPM
BOLD signal
Time =1 2+ +
err
or
x1 x2 e
Single voxel regression model
exxy 2211
Mass-univariate analysis: voxel-wise GLM
=
e+yy XX
N
1
N N
1 1p
pModel is specified by
both1. Design matrix X2. Assumptions about
e
Model is specified by both
1. Design matrix X2. Assumptions about
e
eXy
The design matrix embodies all available knowledge about experimentally controlled factors and
potential confounds.
),0(~ 2INe
N: number of scansp: number of regressors
• one sample t-test• two sample t-test• paired t-test• Analysis of Variance
(ANOVA)• Factorial designs• correlation• linear regression• multiple regression• F-tests• fMRI time series models• Etc..
GLM: mass-univariate parametric analysis
Parameter estimation
eXy
= +
e
2
1
Ordinary least squares estimation
(OLS) (assuming i.i.d. error):
Ordinary least squares estimation
(OLS) (assuming i.i.d. error):
yXXX TT 1)(ˆ
Objective:estimate parameters to minimize
N
tte
1
2
y X
A geometric perspective on the GLM
y
e
Design space defined by X
x1
x2 ˆ Xy
Smallest errors (shortest error vector)when e is orthogonal to X
Ordinary Least Squares (OLS)
0eX T
XXyX TT
0)ˆ( XyX T
yXXX TT 1)(ˆ
x1
x2x2*
y
Correlated and orthogonal regressors
When x2 is orthogonalized w.r.t. x1, only the parameter
estimate for x1 changes, not that for x2!
Correlated regressors = explained variance is
shared between regressors
121
2211
exxy
1;1 *21
*2
*211
exxy
eXy = +
e
2
1
y X
),0(~ 2INe
What are the problems of this model?
What are the problems of this model?
1. BOLD responses have a delayed and dispersed form.
HRF
2. The BOLD signal includes substantial amounts of low-frequency noise (eg due to scanner drift).
3. Due to breathing, heartbeat & unmodeled neuronal activity, the errors are serially correlated. This violates the i.i.d. assumptions of the noise model in the GLM
t
dtgftgf0
)()()(
Problem 1: Shape of BOLD responseSolution: Convolution model
expected BOLD response = input function impulse response function (HRF)
=
Impulses HRF Expected BOLD
Convolution model of the BOLD response
Convolve stimulus function with a canonical hemodynamic response function (HRF):
HRF
t
dtgftgf0
)()()(
blue = data
black = mean + low-frequency drift
green = predicted response, taking into account low-frequency drift
red = predicted response, NOT taking into account low-frequency drift
Problem 2: Low-frequency noise Solution: High pass filtering
discrete cosine transform (DCT) set
discrete cosine transform (DCT) set
discrete cosine transform (DCT) set
High pass filtering
withwithttt aee 1 ),0(~ 2 Nt
1st order autoregressive process: AR(1)
)(eCovautocovariance
function
N
N
Problem 3: Serial correlations
Multiple covariance components
= 1 + 2
Q1 Q2
Estimation of hyperparameters with ReML (Restricted Maximum Likelihood).
V
enhanced noise model at voxel i
error covariance components Qand hyperparameters
jj
ii
QV
VC
2
),0(~ ii CNe
1 1 1ˆ ( )T TX V X X V y
Parameters can then be estimated using Weighted Least Squares (WLS)
Let 1TW W V
1
1
ˆ ( )
ˆ ( )
T T T T
T Ts s s s
X W WX X W Wy
X X X y
Then
where
,s sX WX y Wy
WLS equivalent toOLS on whiteneddata and design
Contrasts &statistical parametric
maps
Q: activation during listening ?
Q: activation during listening ?
c = 1 0 0 0 0 0 0 0 0 0 0
Null hypothesis:Null hypothesis: 01
)ˆ(
ˆ
T
T
cStd
ct
X
Summary
Mass univariate approach.
Fit GLMs with design matrix, X, to data at different points in space to estimate local effect sizes,
GLM is a very general approach
Hemodynamic Response Function
High pass filtering
Temporal autocorrelation
Thank you for your attention!
…and thanks to Guillaume for his slides .
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